In computational and mathematical biology, a biological lattice-gas cellular automaton (BIO-LGCA) is a discrete model for moving and interacting biological agents,^[1] a type of cellular automaton. The BIO-LGCA is based on the lattice-gas cellular automaton (LGCA) model used in fluid dynamics. A BIO-LGCA model describes cells and other motile biological agents as point particles moving on a discrete lattice, thereby interacting with nearby particles. Contrary to classic cellular automaton models, particles in BIO-LGCA are defined by their position and velocity. This allows to model and analyze active fluids and collective migration mediated primarily through changes in momentum, rather than density. BIO-LGCA applications include cancer invasion^[2] and cancer progression.^[3]

Model definition

As are all cellular automaton models, a BIO-LGCA model is defined by a lattice ${\displaystyle {\mathcal {L}}}$, a state space ${\displaystyle {\mathcal {E}}}$, a neighborhood ${\displaystyle {\mathcal {N}}}$, and a rule ${\displaystyle {\mathcal {R}}}$.^[4]

• The lattice (${\displaystyle {\mathcal {L}}}$) defines the set of all possible particle positions. Particles are restricted to occupy only certain positions, typically resulting from a regular and periodic tesselation of space. Mathematically, ${\displaystyle {\mathcal {L}}\subset \mathbb {R} ^{d}}$ is a discrete subset of ${\displaystyle d}$-dimensional space.
• The state space (${\displaystyle {\mathcal {E}}}$) describes the possible states of particles within every lattice site ${\displaystyle \mathbf {r} \in {\mathcal {L}}}$. In BIO-LGCA, multiple particles with different velocities may occupy a single lattice site, as opposed to classic cellular automaton models, where typically only a single cell can reside in every lattice node simultaneously. This makes the state space slightly more complex than that of classic cellular automaton models (see below).
• The neighborhood (${\displaystyle {\mathcal {N}}}$) indicates the subset of lattice sites which determines the dynamics of a given site in the lattice. Particles only interact with other particles within their neighborhood. Boundary conditions must be chosen for neighborhoods of lattice sites at the boundary of finite lattices. Neighborhoods and boundary conditions are identically defined as those for regular cellular automata (see Cellular automaton).
• The rule (${\displaystyle {\mathcal {R}}}$) dictates how particles move, proliferate, or die with time. As every cellular automaton, BIO-LGCA evolves in discrete time steps. In order to simulate the system dynamics, the rule is synchronously applied to every lattice site at every time step. Rule application changes the original state of a lattice site to a new state. The rule depends on the states of lattice sites in the interaction neighborhood of the lattice site to be updated. In BIO-LGCA, the rule is divided into two steps, a probabilistic interaction step followed by a deterministic transport step. The interaction step simulates reorientation, birth, and death processes, and is defined specifically for the modeled process. The transport step translocates particles to neighboring lattice nodes in the direction of their velocities. See below for details.

State space

For modeling particle velocities explicitly, lattice sites are assumed to have a specific substructure. Each lattice site ${\displaystyle \mathbf {r} \in {\mathcal {L}}}$ is connected to its neighboring lattice sites through vectors called "velocity channels", ${\displaystyle \mathbf {c} _{i}}$, ${\displaystyle i\in \{1,2,\ldots ,b\}}$, where the number of velocity channels ${\displaystyle b}$ is equal to the number of nearest neighbors, and thus depends on the lattice geometry (${\displaystyle b=2}$ for a one-dimensional lattice, ${\displaystyle b=6}$ for a two-dimensional hexagonal lattice, and so on). In two dimensions, velocity channels are defined as ${\displaystyle \mathbf {c} _{i}=\left(\cos {\frac {2\pi i}{b}},\sin {\frac {2\pi i}{b}}\right)}$. Additionally, an arbitrary number ${\displaystyle a}$ of so-called "rest channels" may be defined, such that ${\displaystyle \mathbf {c} _{i}=(0,0)}$, ${\displaystyle i\in \{b+1,b+2,\ldots ,b+a\}}$. A channel is said to be occupied if there is a particle in the lattice site with a velocity equal to the velocity channel. The occupation of channel ${\displaystyle \mathbf {c} _{i}}$ is indicated by the occupation number ${\displaystyle s_{i}}$. Typically, particles are assumed to obey an exclusion principle, such that no more than one particle may occupy a single velocity channel at a lattice site simultaneously. In this case, occupation numbers are Boolean variables, i.e. ${\displaystyle s_{i}\in {\mathcal {S}}=\{0,1\}}$, and thus, every site has a maximum carrying capacity ${\displaystyle K=a+b}$. Since the collection of all channel occupation numbers defines the number of particles and their velocities in each lattice site, the vector ${\displaystyle \mathbf {s} =\left(s_{1},s_{2},\ldots ,s_{K}\right)}$ describes the state of a lattice site, and the state space is given by ${\displaystyle {\mathcal {E}}={\mathcal {S}}^{K}}$.

Rule and model dynamics

The states of every site in the lattice are updated synchronously in discrete time steps to simulate the model dynamics. The rule is divided into two steps. The probabilistic interaction step simulates particle interaction, while the deterministic transport step simulates particle movement.

Interaction step

Depending on the specific application, the interaction step may be composed of reaction and/or reorientation operators.

The reaction operator ${\displaystyle {\mathcal {A}}}$ replaces the state of a node ${\displaystyle \mathbf {s} }$ with a new state ${\displaystyle \mathbf {s} ^{\mathcal {A}}}$ following a transition probability ${\displaystyle P\left(\left.\mathbf {s} \rightarrow \mathbf {s} ^{\mathcal {A}}\right|\mathbf {s} _{\mathcal {N}}\right)}$, which depends on the state of the neighboring lattice sites ${\displaystyle \mathbf {s} _{\mathcal {N}}}$to simulate the influence of neighboring particles on the reactive process. The reaction operator does not conserve particle number, thus allowing to simulate birth and death of individuals. The reaction operator's transition probability is usually defined ad hoc form phenomenological observations.

The reorientation operator ${\displaystyle {\mathcal {O}}}$ also replaces a state $$Zdroj:https://en.wikipedia.org?pojem=BIO-LGCA
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